"How old are your three children?" the maths teachers ask a former student. He is told their ages add to 13, and multiply to give the number on his study door (which they can both see). "i will need to know more", the master days, after a few moments relfection. "the eldest one is learning to play the violin", replies his former pupil. "ah! in that case i can now give you their ages", the master tells him, and does so correctly. How does he know? what are their ages? what is the number of the door?

1. list all the possible ages of the children (you don't need to, cos i've done it for you) -

eg. (1, 1, 11) (1, 2, 10) etc
2. if you list all this, you will find that there are 2 answers that give a solution of 36. these are:
(6, 6, 1) and
(9, 2, 2)
3. the master says he needs to know more, because there are solutions from which he must choose, and in order to get the right answer, he needs one more clue.
4. the former student says that "his eldest son...", meaning that in the 3 sons, there is one that is older than the rest.
5. from the 2 options listed:
(6, 6, 1) and (9, 9, 2), we can see that (9, 2, 2) is obviously the answer, because the 9 year old is older than the rset, whereas in the other case, there are two sons with the same age of 6.
6. their ages are 9, 2, and 2, and the number on the door is 36.

The mathematician will only beed to know more if there are two products that are the same (36), 1,6,6, or 2,2,9. In the question it mentions an 'eldest' son, meaning that there is only 1 eldest son (not twins). Therefore their ages are: 2,2,9 and the office number is 36.

To solve this problem, we need to break it down step by step.

Step 1: Determine the possible combinations of ages that add up to 13.
Since we have three children, we can start by listing all the possible combinations of ages that add up to 13:
1 + 1 + 11
1 + 2 + 10
1 + 3 + 9
1 + 4 + 8
1 + 5 + 7
2 + 2 + 9
2 + 3 + 8
2 + 4 + 7
2 + 5 + 6
3 + 3 + 7
3 + 4 + 6
3 + 5 + 5
4 + 4 + 5

Step 2: Consider the additional clue that the children multiply to give the number on the study door.
We need to find the number on the study door, which is the product of the ages of the three children. However, with the information provided, there is no way to solve this step yet. We need more information.

Step 3: Consider the clue that the eldest child is learning to play the violin.
The fact that the eldest child is learning to play the violin tells us that there must be a unique combination of ages that satisfies the problem. This means there is only one possible set of ages.

Let's examine the possible combinations again and see if there is a unique combination:
1 + 1 + 11 --> Not unique.
1 + 2 + 10 --> Not unique.
1 + 3 + 9 --> Not unique.
1 + 4 + 8 --> Not unique.
1 + 5 + 7 --> Not unique.
2 + 2 + 9 --> Not unique.
2 + 3 + 8 --> Not unique.
2 + 4 + 7 --> Not unique.
2 + 5 + 6 --> Not unique.
3 + 3 + 7 --> Not unique.
3 + 4 + 6 --> Unique (Eureka!)
3 + 5 + 5 --> Not unique.
4 + 4 + 5 --> Not unique.

Therefore, the combination of ages must be: 3, 4, and 6.

Step 4: Calculate the number on the study door.
The number on the study door can be found by multiplying the ages of the three children: 3 * 4 * 6 = 72.

Conclusion:
The master deduced that the ages of the three children are 3, 4, and 6. The number on the study door is 72.